Nothing

```
#' Optimization for equi-correlated fixed-X and Gaussian knockoffs
#'
#' This function solves a very simple optimization problem needed to create fixed-X and
#' Gaussian SDP knockoffs on the full the covariance matrix. This may be significantly
#' less powerful than \code{\link{create.solve_sdp}}.
#'
#' @param Sigma positive-definite p-by-p covariance matrix.
#' @return The solution \eqn{s} to the optimization problem defined above.
#'
#' @details Computes the closed-form solution to the semidefinite programming problem:
#' \deqn{ \mathrm{maximize} \; s \quad
#' \mathrm{subject} \; \mathrm{to:} \; 0 \leq s \leq 1, \;
#' 2\Sigma - sI \geq 0 }
#' used to generate equi-correlated knockoffs.
#'
#' The closed form-solution to this problem is \eqn{s = 2\lambda_{\mathrm{min}}(\Sigma) \land 1}.
#'
#' @family optimization
#'
#' @export
create.solve_equi <- function(Sigma) {
# Check that covariance matrix is symmetric
stopifnot(isSymmetric(Sigma))
p = nrow(Sigma)
tol = 1e-10
# Convert the covariance matrix to a correlation matrix
G = cov2cor(Sigma)
# Check that the input matrix is positive-definite
if (!is_posdef(G)) {
stop('The covariance matrix is not positive-definite: cannot solve SDP',immediate.=T)
}
if (p>2) {
converged=FALSE
maxitr=10000
while (!converged) {
lambda_min = RSpectra::eigs(G, 1, which="SR", opts=list(retvec = FALSE, maxitr=100000, tol=1e-8))$values
if (length(lambda_min)==1) {
converged = TRUE
} else {
if (maxitr>1e8) {
warning('In creation of equi-correlated knockoffs, while computing the smallest eigenvalue of the
covariance matrix. RSpectra::eigs did not converge. Giving up and computing full SVD with built-in R function.',immediate.=T)
lambda_min = eigen(G, symmetric=T, only.values = T)$values[p]
converged=TRUE
} else {
warning('In creation of equi-correlated knockoffs, while computing the smallest eigenvalue of the
covariance matrix. RSpectra::eigs did not converge. Trying again with increased number of iterations.',immediate.=T)
maxitr = maxitr*10
}
}
}
} else {
lambda_min = eigen(G, symmetric=T, only.values = T)$values[p]
}
if (lambda_min<0) {
stop('In creation of equi-correlated knockoffs, while computing the smallest eigenvalue of the
covariance matrix. The covariance matrix is not positive-definite.')
}
s = rep(1, nrow(Sigma)) * min(2*lambda_min, 1)
# Compensate for numerical errors (feasibility)
psd = 0;
s_eps = 1e-8;
while (psd==0) {
psd = is_posdef(2*G-diag(s*(1-s_eps),length(s)))
if (!psd) {
s_eps = s_eps*10
}
}
s = s*(1-s_eps)
# Scale back the results for a covariance matrix
return(s*diag(Sigma))
}
```

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